Assignment_2_Solutions

Assignment_2_Solutions - Stat 371 Spring 2011 Assignment 2...

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Stat 371 Spring 2011 Assignment 2 Solutions 1. (Some fun with covariances) Appendix 2 in the course notes deals with expectation, variance and covariance for vectors of random variables. This question will give you some practice playing with the formulas. a) If U and V are two vectors of random variables of length m and n respectively, and is an [ , ] Cov U V m n × matrix with element , prove that th ij [ , ] j i U V j Cov [ , ] [( [ ])( [ ]) ] T Cov U V E U E U V E V = We have and the matrix is m with entry ( and so the result follows. [ , ] [( [ ])( [ ])] i j i i j j Cov U V E U E U V E V = ( [ ])( [ ]) T U E U V E V n × th ij [ ])( [ ]) i i j U E U V E V b) Show that [ , ] [ , ] Cov AU V ACov U V = and [ , ] [ , ] T Cov U BV Cov U V B = Using the results in a), we have [ , ] [( [ ])( [ ]) [ ( [ ])( [ ]) ] [( [ ])( [ ]) ] T T T Cov AU V E AU E AU V E V E A U AE U V E V AE U E U V E V = = = ] and [ , ] [( [ ])( [ ]) ] [( [ ])( [ ]) ] [( [ ]){ ( [ ])} ] [( [ ])( [ ]) ] T T T T T Cov U BV E U E U BV E BV E U E U BV E BV E U E U B V E V E U E U V E V B = = = = c) For the regression model Y X R β = + where 2 ~ (0, ) R N σ I 0 , show that and [ , ] Cov X r β = ± ± [ , ] 0 Cov Y r ± 2 [ , ] [ ,( ) ] [ , ]( ) ( ) 0 T Cov X r Cov HY I H Y HCov Y Y I H H I I H β σ = = = = ± ± since ( ) I H is symmetric. 2 [ , ] [ ,( ) ] [ , ]( ) ( ) T Cov Y r Cov Y I H Y Cov Y Y I H I H σ = = = ± which is not a matrix of 0’s.
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